Let v1 = [1 2 3], v2 = [4 5 6], and v3 = [2 1 0] a. Determine if the set {v1, v2, v3} is linearly independent. b. If possible, find a linear dependence relation among v1 , v2 , and v3.

Question

Let [latex] v_{1} = \left [ \begin{matrix} 1 \ 2 \ 3 \end{matrix} ight ] , v_{2} = \left [ \begin{matrix} 4 \ 5 \ 6 \end{matrix} ight ] , and v_{3} = \left [ \begin{matrix} 2 \ 1 \ 0 \end{matrix} ight ] [/latex]

a. Determine if the set {[latex] v_{1} [/latex], [latex] v_{2} [/latex], [latex] v_{3} [/latex]} is linearly independent.
b. If possible, find a linear dependence relation among [latex] v_{1} [/latex], [latex] v_{2} [/latex], and [latex] v_{3} [/latex].

Accepted Answer

a. We must determine if there is a nontrivial solution of equation (1) above. Row operations on the associated augmented matrix show that

[latex] \left [ \begin{matrix} 1 & 4 & 2 & 0 \ 2 & 5 & 1 & 0 \ 3 & 6 & 0 & 0 \end{matrix} ight ] \sim \left [ \begin{matrix} 1 & 4 & 2 & 0 \ 0 & -3 & -3 & 0 \ 0 & 0 & 0 & 0 \end{matrix} ight ] [/latex]

Clearly, [latex] x_{1} [/latex] and [latex] x_{2} [/latex] are basic variables, and [latex] x_{3} [/latex] is free. Each nonzero value of [latex] x_{3} [/latex] determines a nontrivial solution of (1). Hence [latex] v_{1} [/latex] , [latex] v_{2} [/latex] , [latex] v_{3} [/latex] are linearly dependent (and not linearly independent).

b. To find a linear dependence relation among [latex] v_{1} [/latex] , [latex] v_{2} [/latex] , and [latex] v_{3} [/latex] , completely row reduce the augmented matrix and write the new system:

[latex] \left [ \begin{matrix} 1 & 0 & -2 & 0 \ 0 & 1 & 1 & 0 \ 0 & 0 & 0 & 0 \end{matrix} ight] [/latex] [latex]\begin{matrix} x_{1} & & - 2x_{3} & = 0 \ & x_{2} &+ x_{3} &= 0 \&& 0 & = 0 \end{matrix} [/latex]

Thus [latex] x_{1} [/latex] = 2[latex] x_{3} [/latex], [latex] x_{2} [/latex] = -[latex] x_{3} [/latex], and [latex] x_{3} [/latex] is free. Choose any nonzero value for [latex] x_{3} [/latex]—say, [latex] x_{3} [/latex] = 5. Then [latex] x_{1} [/latex] = 10 and [latex] x_{2} [/latex] = -5. Substitute these values into equation (1) and obtain

[latex] 10v_{1} - 5v_{2} + 5v_{3} = 0 [/latex]

This is one (out of infinitely many) possible linear dependence relations among [latex] v_{1} [/latex], [latex] v_{2} [/latex], and [latex] v_{3} [/latex].

Question 1.7.1: Let v1 = [1 2 3], v2 = [4 5 6], and v3 = [2 1 0] a. Determin...

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